Questions: Expand log(√x). Expand ln(∛(x^2))

Solution Steps

1. For the first question, we need to expand the logarithmic expression \(\log (\sqrt{x})\). We can use the logarithm property \(\log(a^b) = b \log(a)\) to simplify the expression.
2. For the second question, we need to expand the natural logarithmic expression \(\ln \left(\sqrt[3]{x^{2}}\right)\). Similarly, we can use the logarithm property \(\ln(a^b) = b \ln(a)\) to simplify the expression.

Using the property of logarithms, we can expand \(\log(\sqrt{x})\) as follows: \[ \log(\sqrt{x}) = \log(x^{1/2}) = \frac{1}{2} \log(x) \]

Similarly, we can expand \(\ln\left(\sqrt[3]{x^{2}}\right)\) using the same logarithmic property: \[ \ln\left(\sqrt[3]{x^{2}}\right) = \ln\left(x^{2/3}\right) = \frac{2}{3} \ln(x) \]

Final Answer